Frullani integral

In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form
 * $$\int _{0}^{\infty}{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x$$

where $$f$$ is a function defined for all non-negative real numbers that has a limit at $$\infty$$, which we denote by $$f(\infty)$$.

The following formula for their general solution holds if $$f$$ is continuous on $$(0,\infty)$$, has finite limit at $$\infty$$, and $$a,b > 0$$:


 * $$\int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}}\,{\rm {d}}x=\Big(f(\infty)-f(0)\Big)\ln {\frac {a}{b}}.$$

Proof for continuously differentiable functions
A simple proof of the formula (under stronger assumptions than those stated above, namely $$f \in \mathcal{C}^1(0,\infty)$$) can be arrived at by using the Fundamental theorem of calculus to express the integrand as an integral of $$f'(xt) = \frac{\partial }{\partial t} \left(\frac{f(xt)}{x}\right)$$:


 * $$\begin{align}

\frac{f(ax)-f(bx)}{x} &= \left[\frac{f(xt)}{x}\right]_{t=b}^{t=a} \, \\ & = \int_b^a f'(xt) \, dt \\ \end{align}$$

and then use Tonelli’s theorem to interchange the two integrals:


 * $$\begin{align}

\int_0^\infty \frac{f(ax)-f(bx)}{x} \,dx & = \int_0^\infty \int_b^a f'(xt) \, dt \, dx \\ & = \int_b^a \int_0^\infty f'(xt) \, dx \, dt \\ & = \int_b^a \left[\frac{f(xt)}{t}\right]_{x=0}^{x \to \infty}\, dt \\ & = \int_b^a \frac{f(\infty)-f(0)}{t}\, dt \\ & = \Big(f(\infty)-f(0)\Big)\Big(\ln(a)-\ln(b)\Big) \\ & = \Big(f(\infty)-f(0)\Big)\ln\Big(\frac{a}{b}\Big) \\ \end{align}$$ Note that the integral in the second line above has been taken over the interval $$[b,a]$$, not $$[a,b]$$.

Applications
The formula can be used to derive an integral representation for the natural logarithm $$\ln(x)$$ by letting $$f(x) = e^{-x}$$ and $$a=1$$:


 * $${\int _{0}^{\infty}{\frac {e^{-x}-e^{-bx}}{x}}\,{\rm {d}}x=\Big(\lim_{n\to\infty}\frac{1}{e^n}-e^0\Big)\ln \Big({\frac {1}{b}}}\Big) = \ln(b)$$

The formula can also be generalized in several different ways.